1549380232-Automorphic_Forms_and_Applications__Sarnak_

LECTURE 1. FINITE MODELS

b d

-+ - --

-----< -

-----< -----<

• E - K
  -----< -----< ---- ---<
  -----< - - _«
  i -----< - ( <

-----< -- ~

• 
  
  
  
  • ( ~« ___.( ---<
    -----< - - -- ---<
    ~
    
    
    • 
      
      
      
      • ~
      
      
      • 
        
        
        
        • -----< -
          -----<
          -----<
          Po isson Primes n+•Br Sinai Zeros Uniform SL(2 ,Z) finite upper
          ~·> half plane
        
        
        
        
      
      
      
      
    
    
    
    
  
  
  
  

Figure 2. Columns (a)- (f ) are from Bohigas and Giannoni [ 11 ] and column

(g) is from Sarnak [67]. Segments of "spectra,'' each containing 50 levels.

T h e "arrowheads" mark t he occurrence of pairs of levels with spacings smaller

than 1/4. T h e la be ls are explained in the text. Column (h) contains finite

upper half plane graph e igenvalues (wit ho u t multiplicity) for the prime 53,

with o =a= 2.

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(h) should be moved to lie next to column (b ). Column (h) is the sp ectrum of a

finite upper half plane graph for p=53 (a= 6 = 2), without multiplicity.

Exercise 1. Produce your own versions of as many columns of Figure 2 as possible.

A lso make a column for level spacings of lengths of primitive closed geodesics in

SL(2, Z)\H. See pages 277 -28 0 of Terras [83], Vol. I.

Given some favorite operator .C acting on some Hilbert space L^2 (M), as in t he
first paragraph of this section, one h as favorite questions in quantum chaos theory
(see Sarnak [67] or Hejhal et al [37] or the talks given in the spring, 1999 random
matrix theory workshop at t he web site http://www.msri.org)).

Examples o f Q uestio n s of Interest

• 
  
  
  
  1. Does the sp ectrum of .C determine M? Can you hear the shape of M?
  
  
  
  


• 
  
  
  
  2. Give bounds on t he sp ectrum of .C.
  
  
  
  


• 
  
  
  
  3. Is the histogram of the spectrum of .C given by the Wigner semi-circle
     distribution [9 1 ]?
  
  
  
  


• 
  
  
  
  4. Arrange the spectrum of .C so t hat An ::; An+ 1. Consider the level
     spacings JAn+l - An l normalized to h ave mean 1. What is the histogram
     for the level spacings?
  
  
  
  


• 
  
  
  
  5. Behavior of the nodal lines (<Pn = 0) of eigenfunctions of .C as n--> oo.
  
  
  
  

Note that when NI is finite, we will consider a sequence ]Vfj such that mj =

J.Lld'j J --> oo as j --> oo and we seek limit ing distributions of spectra and level

spacings as j --> oo. Now the <Pn = (Jf n), .. ., J$::)) are eigenvectors of the matrix